Octahedron Problem: How Many Color Patterns?

[reese92tsi]

If each side of a tetrahedron is an equilateral triangle painted white or black, five distinct color patterns are possible: all sides white, all black, just one side white and the rest black, just one side black and the rest white, and two sides white while the other two are black. If each side of an octahedron is an equilateral triangle painted white or black, how many distinct patterns are possible?

I was able to come up with a good answer but I'm not sure if its right or not.

 

[chaoseverlasting]

3!.3!.2⁸/2⁶ I think...

The symmetry is the catch here...

 

[tiny-tim]

Welcome to PF!

Hi reese92tsi! Welcome to PF!

I was able to come up with a good answer but I'm not sure if its right or not.

Show us (with your reasoning, of course)!

 

[reese92tsi]

Hi reese92tsi! Welcome to PF!


Show us (with your reasoning, of course)!

Thanks for the warm welcome

all sides black: 1 pattern
all white: 1

7 white 1 black: 1
7 black 1 white: 1

6 white 2 black: 3
6 black 2 white: 3
-2 white faces adjacent at an edge
-2 white faces adjacent at a vertex
-2 white faces neither adjacent at an edge or vertex

5 white 3 black: 3
5 black 3 white: 3
-3 white faces adjacent at edges
-3 white faces adjacent at only vertices
-2 white faces adjacent at an edge and 1 white face
adjacent at either vertex of the set of the 2 edge adjacent white faces

4 black 4 white: 7
-4 white faces sharing a common vertex
-3 white faces sharing a common vertex and 1 white face
adjacent at only the vertex of the rightmost and leftmost
white face of the 3
-3 white faces sharing a common vertex and 1 white face
adjacent to only the edge of the right most face of the 3
-The mirror to the pattern above
-3 white faces sharing a common vertex and 1 white face adjacent to
only the edge of the center white face of the 3
-4 white faces adjacent only at vertices
-2 sets of 2 white faces which are adjacent at an edge
touching each other at only 2 vertices

So I've got 23 patterns

 

[tiny-tim]

5 white 3 black: 3
5 black 3 white: 3
-3 white faces adjacent at edges
-3 white faces adjacent at only vertices
-2 white faces adjacent at an edge and 1 white face
adjacent at either vertex of the set of the 2 edge adjacent white faces

Hi reese92tsi!

0 1 2 and 4 white look good, but I'm not convinced about 3 …

what does 3 white faces adjacent at edges mean?

and have you checked that there are no mirror images left out?

 

[reese92tsi]

I'm sorry about the phrasing. "3 white faces sharing a common vertex" is probably a better way of phrasing it.

Going over the patterns in my head it looks like I've found them all (still not positive).

3!.3!.2⁸/2⁶ I think...

The symmetry is the catch here...

Can you explain this formula in further detail, I'm not really a math buff (would love to become one). I think puzzles/logic problems are a lot of fun.